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Recall that \(\rho = -p\)  is the equation of state for a cosmological constant. The conclusion of Theorem REF  is that \(\widetilde{\rho }(\mathcal {O}) = -\widetilde{p}(\mathcal {O})\) ; this is what we mean by the cosmological constant appears as an initial condition.
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Note that \(M = I^+(\mathcal {O}, M_\text{{\rm ext}})\)  holds for Milne-like spacetimes; see figure REF . Assumption (b) mimics what happens in figure REF . Hence the hypotheses in Theorem REF  generalize what happens in a Milne-like spacetime but without the isotropy assumption.
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Regarding assumption (c), let \((M,g)\)  be a Milne-like spacetime with a scale factor that's analytic at zero: \(a(\tau ) = \tau + \sum _{2}^\infty c_n\tau ^n\) . If \(c_2 \ne 0\) , then it's easy to see from eq. (REF ) that \(\rho \)  and \(p\)  diverge as \(\tau \rightarrow 0\) . So our assumption that \(\rho \)  and \(p\)  extend continuously to \(M \cup \lbrace \mathcal {O}\rbrace \)  is analogous to setting \(c_2 = 0\)  in statement (REF ). Moreover, if \(c_2 = 0\)  and \(c_4 = 0\) , then the Ricci tensor, \(\text{Ric}\) , of \((M,g)\)  extends continuously to \(M \cup \lbrace \mathcal {O}\rbrace \) ; this follows from [1]} since \(\text{Ric}\)  can be written as a sum of products of the metric, its inverse, and their first and second derivatives along with the fact that the inverse metric is as regular as the metric.
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Regarding assumption (d), recall that \((M_\text{{\rm ext}}, g_\text{{\rm ext}})\)  is strongly causal at \(\mathcal {O}\)  means that for any neighborhood \(U\)  of \(\mathcal {O}\)  there is a neighborhood \(V \subset U\)  of \(\mathcal {O}\)  such that \(\gamma (a),\gamma (b) \in V\)  implies \(\gamma \big ([a,b]\big ) \subset U\)  whenever \(\gamma \colon [a,b] \rightarrow M_\text{{\rm ext}}\)  is a future directed causal curve. This assumption holds for the usual continuous extensions of Milne-like spacetimes constructed in section REF  since these constructions are conformal to (subsets of) Minkowski space which are strongly causal at every point.

